A Wiener inversion-type theorem

Holub, James R.

doi:10.1090/S0002-9939-1986-0840618-8

A Wiener inversion-type theorem
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by James R. Holub PDF

Proc. Amer. Math. Soc. 97 (1986), 399-402 Request permission

Abstract:

Let $W(D) = \{ f(z) = \Sigma _{n = 0}^\infty {a_n}{z^n}|\;||f|{|_1} = \Sigma _{n = 0}^\infty |{a_n}| < + \infty \}$, $f(z)$ a function in $W(D)$ for which $f(0) = 1$, and ${M_f}$ the operator of multiplication by $f(z)$ on $W(D)$. It is shown that if $k$ and $m$ are integers for which $0 \leq m \leq k - 1$ and $X_k^m$ is the closed subspace of $W(D)$ spanned by $\{ {z^{nk + i}}|n = 0,1, \ldots ;i = 0,1, \ldots ,m\}$, then ${M_f}$ is bounded below on $X_k^m \Leftrightarrow f(z)$ does not have $k - m$ distinct zeros in any set of the form $\{ {w^i}{z_0}|0 \leq i \leq k - 1;|{z_0}| = 1\}$, where $w$ is a primitive $k$th root of unity.

References

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